maths
Linear Algebra: Eigenvalues, Rank & Cayley-Hamilton
Eigenvalues/eigenvectors, matrix rank, singularity and the Cayley-Hamilton theorem — the fastest-scoring GATE maths topic.
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Key formulas
det(A − λI) = 0 → λ² − tr(A)·λ + det(A) = 0 (2×2)
Characteristic equation: Roots are eigenvalues; verify with trace/sum and determinant/product checks.
A² − tr(A)·A + det(A)·I = 0 (2×2)
Cayley-Hamilton: Substituting A into its own characteristic polynomial; solve for Aⁿ recursively.
rank(A) ≤ min(m, n); rank(AB) ≤ min(rank A, rank B)
Rank rules: Full-rank square matrices are invertible; singular means rank < n.
Cheat sheet
- ▸Eigenvalues come from det(A − λI) = 0. Sum of eigenvalues = trace(A); product = det(A) — instant checks for any numerical answer.
- ▸Eigenvalues of triangular/diagonal matrices are the diagonal entries. Eigenvalues of Aⁿ are λⁿ, of A⁻¹ are 1/λ, of kA are kλ; eigenvectors are unchanged by these transforms.
- ▸A matrix is singular ⟺ det = 0 ⟺ at least one eigenvalue is 0 ⟺ rank < n. Rank = number of non-zero rows in row-echelon form = max independent rows/columns.
- ▸Cayley-Hamilton: every matrix satisfies its own characteristic equation — use it to compute high powers Aⁿ as a combination of I, A, A², …
- ▸Symmetric matrices have real, orthogonal eigenvectors; skew-symmetric have pure imaginary (or zero) eigenvalues; orthogonal matrices have |λ| = 1.
- ▸Number of independent eigenvectors for a repeated eigenvalue may be fewer than its multiplicity (defective matrix) — diagonalisable only when geometric = algebraic multiplicity for every λ.
Eigenvalue properties you can quote
| Matrix | Eigenvalues | Eigenvectors |
|---|---|---|
| Symmetric (real) | All real | Orthogonal basis possible |
| Skew-symmetric | 0 or purely imaginary | Complex pairs |
| Orthogonal | |λ| = 1 | Orthonormal |
| Triangular | Diagonal entries | Standard/simple |
| kA, Aⁿ, A⁻¹ | kλ, λⁿ, 1/λ | Same as A |
Visual
Eigen-idea (2×2, direction not magnitude):
v₁ →│ A·v₁ = λ₁·v₁ (stays on its line)
│
─────────+────────
│
v₂ ↗ A·v₂ rotates it off its line → not an eigenvector
Eigenvectors = directions the matrix only stretches.Top 4 campus interview questions
MCQ mastery quiz
1. The sum of the eigenvalues of a matrix equals its…
2. A matrix is singular if and only if…
3. Eigenvalues of a real symmetric matrix are always…
4. Cayley-Hamilton states that a matrix satisfies…